GATE 2017 EC – Question 17

Networks, Signals and Systems · Sinusoidal Steady State Analysis · 1 mark · Numerical answer

In the circuit shown, the positive angular frequency $\omega$ (in radians per second) at which the magnitude of the phase difference between the voltages $V_1$ and $V_2$ equals $\frac{\pi}{4}$ radians, is ________.

A source $100\cos\omega t$ drives a series loop of a $1\ \Omega$ resistor, a $1$ H inductor and a $1\ \Omega$ resistor. $V_2$ is the voltage across the first resistor and the inductor together, and $V_1$ is the voltage across the second $1\ \Omega$ resistor.

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Correct answer: 1

Explanation

All the elements carry the same current $I$. The voltage $V_1 = I \times 1$ is in phase with $I$, and $V_2 = I(1 + j\omega)$ has the angle $\tan^{-1}\omega$ relative to $I$. The phase difference is $\frac{\pi}{4}$ when $\tan^{-1}\omega = \frac{\pi}{4}$, so $\omega = 1$ rad/s.